The Cauchy problem and the integral operator

Equation () describes a continuous process of reflected wavefield continuation in the time-offset-midpoint domain. In order to find an integral-type operator that performs the one-step offset continuation, I consider the following initial-value (Cauchy) problem for equation ():

Given a post-NMO constant-offset section at half-offset h₁  

$$\left.P(t_n,h,y)\right\vert _{h=h_1}=P^{(0)}_1(t_n,y)$$ (219)

and its first-order derivative with respect to offset  

$$\left.\partial P(t_n,h,y)\over \partial h\right\vert _{h=h_1}=P^{(1)}_1(t_n,y)\;,$$ (220)

find the corresponding section P⁽⁰⁾(t_n,y) at offset h.

Equation () belongs to the hyperbolic type, with the offset coordinate h being a ``time-like'' variable and the midpoint coordinate y and the time t<sub>n</sub> being ``space-like'' variables. The last condition () is required for the initial value problem to be well-posed Courant (1962). From a physical point of view, its role is to separate the two different wave-like processes embedded in equation (), which are analogous to inward and outward wave propagation. We will associate the first process with continuation to a larger offset and the second one with continuation to a smaller offset. Though the offset derivatives of data are not measured in practice, they can be estimated from the data at neighboring offsets by a finite-difference approximation. Selecting a propagation branch explicitly, for example by considering the high-frequency asymptotics of the continuation operators, can allow us to eliminate the need for condition (). In this section, I discuss the exact integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem (-) for equation () is obtained in Appendix  with the help of the classic methods of mathematical physics. It takes the explicit form

$$\begin{eqnarray} P(t_n,h,y) & = & \int\!\!\int P^{(0)}_1(t_1,y_1)\,G_0(t_1,h_1,y... ...\!\int P^{(1)}_1(t_1,y_1)\,G_1(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\;,\end{eqnarray}$$ (221)

where the Green's functions G₀ and G₁ are expressed as

$$\begin{eqnarray} G_0(t_1,h_1,y_1;t_n,h,y) & = & \mbox{sign}(h-h_1)\,{H(t_n) \ove... ..._n \over t_1^2}\,\left\{ H(\Theta) \over \sqrt{\Theta}\right\}\;,\end{eqnarray}$$ (222) (223)

and the parameter $\Theta$ is  

$$\Theta(t_1,h_1,y_1;t_n,h,y) = \left(h_1^2/t_1^2-h^2/t_n^2\right)\,\left(t_1^2-t_n^2\right)- \left(y_1-y\right)^2\;.$$ (224)

H stands for the Heavyside step-function.

From equations () and () one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality  

$$\Theta(t_1,h_1,y_1;t_n,h,y) = 0\;,$$ (225)

which describes the ``wavefronts'' of the offset continuation process. In terms of the theory of characteristics Courant (1962), the surface $\Theta=0$ corresponds to the characteristic conoid formed by the bi-characteristics of equation () - time rays emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$. The common-offset slices of the characteristic conoid are shown in the left plot of Figure .

 

con
Figure 7 Constant-offset sections of the characteristic conoid - ``offset continuation fronts'' (left), and branches of the conoid used in the integral OC operator (right). The upper part of the plots (small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.

As a second-order differential equation of the hyperbolic type, equation () describes two different processes. The first process is ``forward'' continuation from smaller to larger offsets, the second one is ``reverse'' continuation in the opposite direction. These two processes are clearly separated in the high-frequency asymptotics of operator (). To obtain the asymptotical representation, it is sufficient to note that ${1 \over \sqrt{\pi}}\, {H(t) \over \sqrt{t}}$ is the impulse response of the causal half-order integration operator and that $H(t^2-a^2) \over \sqrt{t^2-a^2}$ is asymptotically equivalent to $H(t-a) \over {\sqrt{2a}\,\sqrt{t-a}}$ (t, a >0). Thus, the asymptotical form of the integral offset-continuation operator becomes

$$\begin{eqnarray} P^{(\pm)}(t_n,h,y) & = & {\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm... ...h,t_n)\, P^{(1)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;.\end{eqnarray}$$ (226)

Here the signs ``+'' and ``-'' correspond to the type of continuation (the sign of h-h₁), ${\bf D}^{1/2}_{\pm\,t_n}$ and ${\bf I}^{1/2}_{\pm\,t_n}$ stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable t_n, the summation paths $\theta^{(\pm)}(\xi;h_1,h,t_n)$ correspond to the two non-negative sections of the characteristic conoid () (Figure ):

 

$$t_1=\theta^{(\pm)}(\xi;h_1,h,t_n)= {t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,$$ (227)

where $U=h^2+h_1^2-\xi^2$, and $V=\sqrt{U^2-4\,h^2\,h_1^2}$; $\xi$ is the midpoint separation (the integration parameter), and $w^{(\pm)}_0$and $w^{(\pm)}_1$ are the following weighting functions:

$$\begin{eqnarray} w^{(\pm)}_0 & = & {1 \over \sqrt{2\,\pi}}\, {\theta^{(\pm)}(\xi... ...rt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;.\end{eqnarray}$$ (228) (229)

Expression () for the summation path of the OC operator was obtained previously by Stovas and Fomel (1993, 1996) and Biondi and Chemingui (1994a,b). A somewhat different form of it is proposed by Bagaini and Spagnolini (1996). I describe the kinematic interpretation of formula () in Appendix .

In the high-frequency asymptotics, it is possible to replace the two terms in equation () with a single term Fomel (1996b). The single-term expression is

$$P^{(\pm)}(t_n,h,y) = {\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm... ..._n)\, P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;,$$ (230)

where

$$\begin{eqnarray} w^{(+)} & = & \sqrt{\theta^{(+)}(\xi;h_1,h,t_n) \over {2\,\pi}}... ...\over \sqrt{2\,\pi t_n}}\; {{h_1^2-h^2 +\xi^2} \over {V^{3/2}}}\;.\end{eqnarray}$$ (231) (232)

A more general approach to true-amplitude asymptotic offset continuation is developed by Santos et al. (1997).

The limit of expression () for the output offset h approaching zero can be evaluated by L'Hospitale's rule. As one would expect, it coincides with the well-known expression for the summation path of the integral DMO operator Deregowski and Rocca (1981)  

$$t_1=\theta^{(-)}(\xi;h_1,0,t_n)= \lim_{h \rightarrow 0} {{t_... ...rt{{U - V} \over 2 }}= {{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.$$ (233)

I discuss the connection between offset continuation and DMO in the next section.